Question 1

Which of the following is a parametric representation of the given surface?   [Assume the axes are oriented as shown in the figure below.]

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 2

Find an equation in x, y, and z for the plane defined below.

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 3

Determine all values of t at which the given vector-valued function is continuous.

Accepted Answer

A)    
B)    
C)    
D)  all real numbers 
A)    
B)    
C)    
D)  all real numbers

Question 4

Evaluate the given integral.

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 5

Find the position function from the given velocity function.

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 6

Find the unit tangent vector to the curve at the indicated point.

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 7

Find the curvature of the polar curve at the indicated point.

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 8

Find the unit normal vector at the given point.

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 9

Evaluate the given integral.

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 10

Use the graph of   below to predict whether the curvature would be larger at   or   .

Accepted Answer

A)    
B)    
C)    
A)    
B)    
C)

Question 11

Find the binormal vector   at t = 2.

Accepted Answer

A)    
B)    
C)    
D)  The binormal vector is undefined at this point. 
A)    
B)    
C)    
D)  The binormal vector is undefined at this point.

Question 12

Graph the curvature function   for   and find the limit of the curvature as   .

Accepted Answer

A)      
B)      
C)      
D)      
A)      
B)      
C)      
D)

Question 13

Find the unit normal vector at the given point.

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 14

Consider the claim that there exists a curve   for which   . Do you agree with this claim? Explain or justify your answer. [You need not give a rigorous proof.]

Accepted Answer

The claim cannot be true. A curve with a

Question 15

Find t such that   and   are perpendicular.

Accepted Answer

A)    
B)  t = 0 
C)    
D)  The vectors   and   are not perpendicular for any value of t. 
A)    
B)  t = 0 
C)    
D)  The vectors   and   are not perpendicular for any value of t.

Question 16

Phillip claims that the trajectory   for   is identical to the trajectory   for   , where t represents time. Sharon suggests that Phillip's claim is not necessarily true. Who is correct and why?

Accepted Answer

Sharon is correct. While the curves trac

Question 17

Show that there are no values of t such that   and   are parallel. Show your work.

Accepted Answer

Answers may vary.
The vectors

Question 18

Use graphing technology to sketch the curve traced out by the given vector-valued function.   Use the viewpoint   and   (see figure below).

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 19

Graph the curvature function   for   and find the limit of the curvature as   .

Accepted Answer

A)      
B)      
C)      
D)      
A)      
B)      
C)      
D)

Question 20

Find all values of t such that   is parallel to the xy-plane.

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Which of the following is a parametric representation of the given surface? [Assume the axes are oriented as shown in the figure below.]

Find an equation in x, y, and z for the plane defined below.

Determine all values of t at which the given vector-valued function is continuous.

Evaluate the given integral.

Find the position function from the given velocity function.

Find the unit tangent vector to the curve at the indicated point.

Find the curvature of the polar curve at the indicated point.

Find the unit normal vector at the given point.

Evaluate the given integral.

Use the graph of below to predict whether the curvature would be larger at or .

Find the binormal vector at t = 2.

Graph the curvature function for and find the limit of the curvature as .

Find the unit normal vector at the given point.

Consider the claim that there exists a curve for which . Do you agree with this claim? Explain or justify your answer. [You need not give a rigorous proof.]

Find t such that and are perpendicular.

Phillip claims that the trajectory for is identical to the trajectory for , where t represents time. Sharon suggests that Phillip's claim is not necessarily true. Who is correct and why?

Show that there are no values of t such that and are parallel. Show your work.

Use graphing technology to sketch the curve traced out by the given vector-valued function. Use the viewpoint and (see figure below).

Graph the curvature function for and find the limit of the curvature as .

Find all values of t such that is parallel to the xy-plane.

Preliminaries

Limits and Continuity

Differentiation

Applications of the Derivative

Integration

Applications of the Definite Integral

Exponentials, Logarithms and Other Transcendental Functions

Integration Techniques

First-Order Differential Equations

Infinite Series

Parametric Equations and Polar Coordinates

Vectors and the Geometry of Space

Functions of Several Variables and Partial Differentiation

Multiple Integrals

Vector Calculus

Second Order Differential Equations

Filters

Exam 13: Vector-Valued Functions

Which of the following is a parametric representation of the given surface? [Assume the axes are oriented as shown in the figure below.]

Find an equation in x, y, and z for the plane defined below.

Determine all values of t at which the given vector-valued function is continuous.

Evaluate the given integral.

Find the position function from the given velocity function.

Find the unit tangent vector to the curve at the indicated point.

Find the curvature of the polar curve at the indicated point.

Find the unit normal vector at the given point.

Evaluate the given integral.

Use the graph of below to predict whether the curvature would be larger at or .

Find the binormal vector at t = 2.

Graph the curvature function for and find the limit of the curvature as .

Find the unit normal vector at the given point.

Consider the claim that there exists a curve for which . Do you agree with this claim? Explain or justify your answer. [You need not give a rigorous proof.]

Find t such that and are perpendicular.

Phillip claims that the trajectory for is identical to the trajectory for , where t represents time. Sharon suggests that Phillip's claim is not necessarily true. Who is correct and why?

Show that there are no values of t such that and are parallel. Show your work.

Use graphing technology to sketch the curve traced out by the given vector-valued function. Use the viewpoint and (see figure below).

Graph the curvature function for and find the limit of the curvature as .

Find all values of t such that is parallel to the xy-plane.

Preliminaries

Limits and Continuity

Differentiation

Applications of the Derivative

Integration

Applications of the Definite Integral

Exponentials, Logarithms and Other Transcendental Functions

Integration Techniques

First-Order Differential Equations

Infinite Series

Parametric Equations and Polar Coordinates

Vectors and the Geometry of Space

Functions of Several Variables and Partial Differentiation

Multiple Integrals

Vector Calculus

Second Order Differential Equations

Filters